(CNN) — After showing off their royal bundle to the world, the Duke and Duchess of Cambridge are doing what new parents do after the big birth show is over. They enjoy some incredibly sweet moments alone with their baby.

Kensington Palace made an official statement about this development Wednesday, saying that William and Catherine are now down to the business of getting “to know their son.”

British media are feverishly reporting where they might be, saying that the couple and child are in the village of Bucklebury, where Catherine’s parents live.

Earlier Wednesday, Queen Elizabeth II went to Kensington Palace to see the newborn for the first time. Uncle Harry paid a visit, too, the palace told reporters.

The parents and the as-yet-named, perhaps future King left the palace by car around lunchtime, UK media reported. The couple has been mum about their immediate schedules.

Tuesday evening, though, they brought their baby out of the Lindo Wing of St. Mary’s Hospital in London, delighting throngs of journalists camped out for a first peek.

Anticipation remains fevered, of course, because the world doesn’t know what to call the new royal.

The prince and his wife are “still working on a name,” William said on the hospital steps, “so we’ll have that as soon as we can — it’s the first time we’ve seen him really, so we’re having a proper chance to catch up.”

Many bets are being placed as the wait continues for the baby’s name to be announced. British bookmakers Ladbrokes have George and James as favorites Wednesday, followed by Alexander, Arthur, Louis and Henry.

William’s name was announced a few days after birth; his brother Harry’s upon leaving the hospital.

The baby has a “good pair of lungs,” the prince told well-wishers. “He’s got her looks, thankfully.”

They took turns holding the child, wrapped in a cream-colored blanket. The parents waved.

William said he’d already changed his first diaper.

“It’s very emotional. It’s such a special time,” Catherine said.

Third in line

The 8-pound, 6-ounce boy was born Monday afternoon. He’s third in line, behind Charles and William, to the British throne.

As well as ruling the United Kingdom, the boy could one day be king of 15 other commonwealth countries that have the British monarch as head of state if none change their constitution in the meantime.

They include Australia, Canada, New Zealand, Belize and Jamaica.

On their way out, the couple walked down the same hospital steps where Diana, Princess of Wales, and Prince Charles gave the world its first look at Prince William 31 years ago.

William placed the teeny royal heir in a car seat in the back of a black sport utility vehicle, then got behind the wheel for the trip to their residence at Kensington Palace in London.

The grand apartment they will eventually move into within the palace, Apartment 1A, is still being refurbished, so William and Catherine have been living in a small cottage in the grounds.

The internal renovation work at Kensington Palace is due to be completed in the fall. The duke and duchess’ staff will also move into refurbished offices there, according to Buckingham Palace accounts released last month.

The late Diana moved into Kensington Palace after marrying Prince Charles in 1981. William and Harry were raised there.

When she died in 1997, mourners laid flowers and tributes outside the palace gates.

‘Absolutely beautiful’

On Tuesday, London reverberated with the sound of cannon fire and peals of bells to mark the birth.

Shortly before the new baby’s departure from St. Mary’s, Prince Charles stopped by for a brief visit with his first grandchild, accompanied by his wife, Camilla, the Duchess of Cornwall. He told reporters it was “marvelous.”

And Catherine’s parents, Carole and Michael Middleton, visited earlier. As grandmothers tend to do, she described her grandbaby as “absolutely beautiful.”

She and her husband were “so thrilled” about being grandparents.

“It was so exciting. It was fantastic,” said Eliza Wells, one of the well-wishers gathered outside the hospital. “The crowd erupted, because everyone’s been waiting so long for it.”

William and Catherine “both seemed very relaxed, even with the press there and the crowd,” Wells said. “They just seemed like a normal couple.”

A normal life?

Royal commentators said the Duke and Duchess of Cambridge will try to give their son as regular an upbringing as possible.

But the intense media interest in the birth of the new prince highlights the challenge his parents face in trying to protect his privacy and maintain a degree of normalcy.

“This baby has two things stopping it from being normal,” historian Kate Williams said. “No. 1, it lives in a life of incredible wealth and privilege. … No. 2, it is an incredible celebrity, and we’ve seen this with the coverage.”

But Prince William loved that his mother tried to give him as normal a childhood as possible, including trips to the cinema and an amusement park, and sending him to a local private school as a boy. “And that’s what he wants for little baby Cambridge,” Williams offered.

Although the excitement over his birth is not universal, there’s no doubting the level of global interest in the prince.

On Monday, there were more than 19 million Facebook interactions related to the royal baby, according to the site.